# Rank and order of a finite group admitting a Frobenius group of automorphisms

Khukhro, E. I. (2013) Rank and order of a finite group admitting a Frobenius group of automorphisms. [MIMS Preprint]

Suppose that a finite group $G$ admits a Frobenius group of automorphisms $FH$ of coprime order with kernel $F$ and complement $H$. In the case where $G$ is a finite $p$-group such that $G=[G,F]$ it is proved that the order of $G$ is bounded above in terms of the order of $H$ and the order of the fixed-point subgroup $C_G(H)$ of the complement, and the rank of $G$ is bounded above in terms of $|H|$ and the rank of $C_G(H)$. Earlier such results were known under the stronger assumption that the kernel $F$ acts on $G$ fixed-point-freely. As a corollary, in the case where $G$ is an arbitrary finite group with a Frobenius group of automorphisms $FH$ of coprime order with kernel $F$ and complement $H$, estimates are obtained of the form $|G|\leq |C_G(F)|\cdot f(|H|, |C_G(H)|)$ for the order, and ${\bf r}(G)\leq {\bf r}(C_G(F))+ g(|H|, {\bf r}(C_G(H)))$ for the rank, where $f$ and $g$ are some functions of two variables.